Abstract

We consider a Hodgkin–Huxley-type model of oscillatory activity in neurons of the snail Helix pomatia. This model has a distinctive feature: It demonstrates multistability in oscillatory and silent modes that is typical for the thalamocortical neurons. A single neuron cell can demonstrate a variety of oscillatory activity: Regular and chaotic spiking and bursting behavior. We study collective phenomena in small and large arrays of nonidentical cells coupled by models of electrical and chemical synapses. Two single elements coupled by electrical coupling show different types of synchronous behavior, in particular in-phase and antiphase synchronous regimes. In an ensemble of three inhibitory synaptically coupled elements, the phenomenon of sequential synchronous dynamics is observed. We study the synchronization phenomena in the chain of nonidentical neurons at different oscillatory behavior coupled with electrical and chemical synapses. Various regimes of phase synchronization are observed: (i) Synchronous regular and chaotic spiking; (ii) synchronous regular and chaotic bursting; and (iii) synchronous regular and chaotic bursting with different numbers of spikes inside the bursts. We detect and study the effect of collective synchronous burst generation due to the cluster formation and the oscillatory death.

Lead Paragraph: Modeling of dynamics of neural systems is important because such a consideration may clarify and resolve different problems that are unsolvable with other methods. We use the dynamical systems theory approach in our study. The model is represented by a system of ordinary differential equations in Cauchy form. It is a complicated analogue of the Hodgkin–Huxley model, which describes the ionic currents caused by diffusion of ions through the cell’s membrane and changing of the membrane potential. We investigate the steady states, bifurcation from excitable to oscillatory mode, and other possible regimes of activity in a single modelneuron. It has been shown that the model is multistable and it reproduces a wide set of activities observable in real biological neurons. We consider networks of nonidenticalneurons connected with electrical and chemical synaptic coupling. Such models demonstrate several interesting effects that may be observed in biological experiments. First, we look into different types of phase synchronization which arise at different parameters of the model and the coupling strength. It was found that the neurons can exhibit in-phase and antiphase synchronization and switching from one type of synchronous behavior to another. We also consider the effects of network burst generation and the effect of the oscillation death in the neuronal chain. We also focus on the effect of firing sequence generation by a motif of inhibitory coupled neurons. Sensitivity of sequential switching dynamics to the initial conditions is shown.